How can we focus students on the various data types in multivariable calculus?

Question

To try to find out if students knew what the gradient was, after the computational questions, I asked the following question on an exam:

Let $f(x, y) = 5 - x - y$. Why doesn't it make sense to find $\nabla(\nabla f)$?

My students almost universally failed to give a good answer to the question. The most common answers are things like

• $\nabla f$ already points in the direction of steepest ascent, so doing it again gives you the same thing.

• Perpendicular to $\nabla f$ is a level curve.

These kind of answers show a lack of precise knowledge of what kind of object the gradient is. This kind of "data type understanding" is at the core of a lot of new student problems in vector calculus.

A correct answer should be something like

• $\nabla f$ is a vector; it doesn't make sense to find the gradient of a vector.

---

I'm interested in your approaches to help students focus on the different data types involved in multivariable calculus (some objects are vectors, some are scalars, many operations turn vectors into scalars or pairs of vectors into one or the other). Would you recommend writing things like:

$\nabla : $ scalar-valued functions of $n$ variables $\to$ $n$-dimensional vectors

when we introduce new objects in this course? That's pretty complicated. Or something else? What can I do to better emphasize data types?

Answer 1

I might seem picky, but I would first refrain from saying that $\nabla f$ is a vector. It is a vector field. This might be considered a common abuse of vocabulary, but using it amounts to assuming that student can fix it up routinely. The problem you are faced with shows very convincingly they don't.

I bet we all have been confronted to someone who, asked to compute the derivative of say $\cos^2x$ at $x=\pi$, wrote $$(\cos^2(\pi))'=((-1)^2)'=(1)'=0$$ Even if your students seem to have passed this level of confusion once they reached the multivariate calculus class, they very probably can fall in it again when confronted with the more sophisticated material of this class. Therefore you need much greater care and rigor than will be strictly necessary later on, with students who passed the class.

Now to the heart of your problem. I don't pretend I have a complete solution, it is a prevalent and very difficult problem; my propositions are partial, and are what I try to do (I do not follow them as strictly as I should myself, even if I speak first person below). I would advise to

1. discuss the question extensively in class,
2. regularly make quizzes on the nature of objects,
3. make clear that the understanding of the nature of the object will be tested.

1. discuss the question extensively in class

Multivariate calculus is probably one of the most ill-understood by student at the point of curriculum it is considered. It is quite easy to get students able to make a number of computations, but very difficult to get them to understand the meaning of the computed objects. One should constantly discuss the natural set to which belongs each object: $f$, $\nabla f$, $df$, $\nabla f(x_1,\dots,x_n)$, $Df(x_1,\dots,x_n)$, $Df(x_1,\dots,x_n)\bullet(v_1,\dots,v_n)$, etc. Doing so at first is in my experience not sufficient, one has to discuss this at each lecture.

One should also discuss in class the content of your question. Present a vector field and discuss whether it makes sense or not to take its gradient. Explain why it does make sense to write $\nabla\big(\langle \nabla f, \nabla g\rangle \big)$; everything that gives you opportunities to discuss the nature of more or less complex objects.

2. regularly make quizzes on the nature of objects

Student need to be tested regularly, to assess their level of understanding in order to adjust the lecture, but also to convince them to work the s*** out of this mess. It is sometimes ungrateful work, but it is important work if multivariate calculus is to be taught at all (and rewarding once one gets it).

Some quizzes should closely match what has been discussed in class, some can be given in advance to show student they did not understand what they thought they did, to introduce some explanations they might be reluctant to give attention to at first. For the anecdote, here is my favorite kind of question:

Consider $F:\mathbb{R}^3\to\mathbb{R}$ defined by $F(x,y,z)=xyz-x^2-y^2-z^2$

1. Compute the differential $DF$ of $F$.
2. Compute $DF(1,2,3)\bullet(4,5,6)$, i.e. the differential of $F$ at the point of coordinates $(1,2,3)$, evaluated on the vector of coordinates $(4,5,6)$.

I usually have 100% correct answers to 1. but no correct answer to 2. To make my point clear I grade this two questions together, with no partial credit if 2. does not have a correct answer. After nailing my students with this, I can expect them to answer similar questions correctly on the final test. It took me some time assuming they would answer this correctly before realizing I had to ask them, and ask them hard.

3. Make clear that the understanding of the nature of the object will be tested.

There is a more general point here. Students have a way of finding out when we are bluffing. Each time we mention something is really important, but it seems quite difficult to them, and we don't test it for fear they will fail, they register the event and realize even more that we don't really mean it when we say it is important. I thus try not to say something is important if I don't want to test it, and I try to test everything that I mentioned as important. If I don't have the time to test everything, I give priority with what students would not want to be on the test. The goal is not to make them miserable, so I tell them in advance how I work for them to be warned. They usually think I BS them, until one or two tests go by, and then they know I am serious and start taking me (and, more importantly, their need to work) seriously.

Answer 2

This is an excellent and important question! After a few years away from it, I'm teaching Multivariate Calculus this semester again, and I'm struggling with the same issues. (Well, don't we all?) The answer above from @BenoîtKloeckner is absolutely amazing, but I'd like to add one more thing. In fact, this comes straight from my experience teaching this course this semester.

The thing is that I found myself talking about data types from day one. In Multivariate Calculus this becomes an issue to be addressed from the very beginning — as soon as we have both vectors and scalars around. Actually, even earlier on, there's inevitably an opportunity — and I use it! — to talk about "data types" when students set up the vector between points $P$ and $Q$ as "$\overrightarrow{PQ}=P-Q$". This immediately prompts a dialogue like:

– Why did you take off points here?

– What are $P$ and $Q$? What kinds of objects are they?

– They are points.

– And is subtraction an operation on points? Did we ever define subtraction of points, or did we define subtraction for something else?

– Oh…

And then next we get into dot and cross products, where emphasizing the types of objects should be a daily routine. So my answer to your question is: Yes, of course it's extremely important, but in Multivariate Calculus this should be emphasized from the very beginning of the semester in a variety of contexts, so that by the time you get to gradients students already know that understanding data types is a big deal (and gradients is just yet another context where it's again important).

In fact, even in earlier classes there are some opportunities for emphasizing data types. For example, one of the common answers to the question thrown into the audience in Calculus I: "How do we interpret geometrically the derivative at a point?", is that "It is the tangent line", to which I retort that "No, it's not. The tangent line is a geometrical object, while the derivative at a point is a number, a different kind of object. It is related to the tangent line, but it is not the tangent line." But if in earlier courses I find myself addressing this only sporadically, in Multivariate Calculus it is a really big deal!

Answer 3

Two things stand out about your question "Why doesn't it make sense to find $\nabla \left(\nabla f\right)$?":

1. What you're really doing is testing whether students understand the domain of the map $\nabla$. I assume you would spend lots of time in a college algebra class, testing their understanding of the domain of a function of a single variable, whether through context ("x represents the number of years after 1990") or admissible input values ("x cannot be 3"). In your problem, the data type is the issue, and this may be a new concept to explore for your class. This could be the time to add this to the "list of ways a relationship may fail". You may be introducing this issue for the first time in this course, so you could couch the data-type issue in terms of what they've already done with the domain of a real-valued function.
2. Problems of this type require practice, so it may have caught some of them off guard if they hadn't been prepped to think along those lines. I think students sometimes read the same meaning into the phrases "Why doesn't it make sense to..." and "Why can't you...", where I might have meant "redundant" in the first case and "impossible" in the second. Those subtle differences tend to skew where their minds go, when all I wanted to know from them was whether they knew that "$\nabla$ doesn't act on vectors", or whatever. Thought: What if you had given them a list of options for the "best reason" and let them choose? Maybe giving multiple choice practice problems a la "choose the best reason" would relieve them of having to come up with perfect wording, but tell you who knew what was really going on.

As for writing things like $\nabla: f(x_1,x_2,\cdots,x_n) \mapsto \vec{x}$, I don't think this is necessarily too complicated (with practice). Vector calculus students are sophisticated enough to handle the new "larger" idea and notation about data types. Isn't it common practice for Calc books to say things like: $\frac{d}{dx}:\text{Quadratic Polynomials} \rightarrow \text{Linear Polynomials}$?

Answer 4

Give a regular quiz similar to this one. Add questions as new data types become available.

Let $f : \mathbb{R}^2 \to \mathbb{R}$, $g: \mathbb{R} \to \mathbb{R}^2$, and $h: \mathbb{R}^2 \to \mathbb{R}^2$.

For each of the following expressions, indicate which of the following options best represents the type of the expression:

• expression does not make sense
• real number
• vector
• vector field
• function of type $\mathbb{R} \to \mathbb{R}$
• function of type $\mathbb{R}^2 \to \mathbb{R}$
• function of type $\mathbb{R} \to \mathbb{R}^2$

1. $f(2,3)$
2. $f(2,t)$
3. $f(2)$
4. $g(2,3)$
5. $g(2)$
6. $g \circ h$
7. $h \circ g$
8. $g(1)\cdot f(1,1)$
9. $g'(1) \cdot \nabla f\big|_{(1,1)}$
10. $\nabla f\big|_{(1,2)} \cdot \langle 4, 5\rangle$
11. $Dh(g'(t))$
12. $Df(g(t))$
13. $Dg(f(t))$

Answer 5

I think the panapoly of curves, vectors and surfaces is what makes 3rd semester calculus a bit of a mine field. There is a reason why there was a book written called "Div, Grad, Curl, and All That."

Maybe it is just a little bit like QM, where you just have to get used to the strangeness (single variable calculus on the other hand seems very intuitive).

So in addition to your test question, I would just show some sort of table that has the different terms and concepts and then which are vectors or curves or what have you. You can have the formula in the table, the name, the notation, what it is useful for (application), physical analogy, other?.

I seem to remember such a table used by Feynman in his lectures for exactly this topic.

Show the table, drill on it. Occasionally ask questions on it. I think this is the only way to get used to Div, Grad, Curl and All That.